Abstract

We investigate the lasing modes in large-Fresnel-number laser systems with astigmatism effects. Experimental results reveal that numerous lasing modes are concentrated on exotic patterns corresponding to intriguing geometries. We theoretically use the quantum operator algebra to construct the wave representation for manifesting the origin of the localized wave patterns.

© 2010 Optical Society of America

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  1. M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, Phys. Rev. A 43, 5090 (1991).
    [CrossRef] [PubMed]
  2. D. Dangoisse, D. Hennequin, C. Lepers, E. Louvergneaux, and P. Glorieux, Phys. Rev. A 46, 5955 (1992).
    [CrossRef] [PubMed]
  3. E. Cabrera, O. G. Calderón, S. Melle, and J. M. Guerra, Phys. Rev. A 73, 053820 (2006).
    [CrossRef]
  4. K. F. Huang, Y. F. Chen, H. C. Lai, and Y. P. Lan, Phys. Rev. Lett. 89, 224102 (2002).
    [CrossRef] [PubMed]
  5. E. G. Abramochkin and V. G. Volostnikov, J. Opt. A 6, S157 (2004).
    [CrossRef]
  6. L. E. Vicent and K. B. Wolf, J. Opt. Soc. Am. A 25, 1875 (2008).
    [CrossRef]
  7. Y. F. Chen, T. H. Lu, K. W. Su, and K. F. Huang, Phys. Rev. Lett. 96, 213902 (2006).
    [CrossRef] [PubMed]
  8. T. H. Lu, Y. C. Lin, Y. F. Chen, and K. F. Huang, Phys. Rev. Lett. 101, 233901 (2008).
    [CrossRef] [PubMed]
  9. M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, Phys. Rev. A 56, 4064 (1997).
    [CrossRef]
  10. S. J. Van Enk and G. Nienhuis, J. Mod. Opt. 41, 963 (1994).
    [CrossRef]
  11. G. Nienhuis and L. Allen, Phys. Rev. A 48, 656 (1993).
    [CrossRef] [PubMed]
  12. S. J. van Enk and G. Nienhuis, Opt. Commun. 94, 147 (1992).
    [CrossRef]
  13. G. Nienhuis and J. Visser, J. Opt. A 6, 248 (2004).
    [CrossRef]
  14. J. Schwinger, in Quantum Theory of Angular Momentum, L.C.Biedenharn and H.van Dam, eds. (Academic, 1965), pp. 229-279.
  15. J. Visser and G. Nienhuis, Phys. Rev. A 70, 013809 (2004).
    [CrossRef]

2008 (2)

T. H. Lu, Y. C. Lin, Y. F. Chen, and K. F. Huang, Phys. Rev. Lett. 101, 233901 (2008).
[CrossRef] [PubMed]

L. E. Vicent and K. B. Wolf, J. Opt. Soc. Am. A 25, 1875 (2008).
[CrossRef]

2006 (2)

Y. F. Chen, T. H. Lu, K. W. Su, and K. F. Huang, Phys. Rev. Lett. 96, 213902 (2006).
[CrossRef] [PubMed]

E. Cabrera, O. G. Calderón, S. Melle, and J. M. Guerra, Phys. Rev. A 73, 053820 (2006).
[CrossRef]

2004 (3)

G. Nienhuis and J. Visser, J. Opt. A 6, 248 (2004).
[CrossRef]

J. Visser and G. Nienhuis, Phys. Rev. A 70, 013809 (2004).
[CrossRef]

E. G. Abramochkin and V. G. Volostnikov, J. Opt. A 6, S157 (2004).
[CrossRef]

2002 (1)

K. F. Huang, Y. F. Chen, H. C. Lai, and Y. P. Lan, Phys. Rev. Lett. 89, 224102 (2002).
[CrossRef] [PubMed]

1997 (1)

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, Phys. Rev. A 56, 4064 (1997).
[CrossRef]

1994 (1)

S. J. Van Enk and G. Nienhuis, J. Mod. Opt. 41, 963 (1994).
[CrossRef]

1993 (1)

G. Nienhuis and L. Allen, Phys. Rev. A 48, 656 (1993).
[CrossRef] [PubMed]

1992 (2)

S. J. van Enk and G. Nienhuis, Opt. Commun. 94, 147 (1992).
[CrossRef]

D. Dangoisse, D. Hennequin, C. Lepers, E. Louvergneaux, and P. Glorieux, Phys. Rev. A 46, 5955 (1992).
[CrossRef] [PubMed]

1991 (1)

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, Phys. Rev. A 43, 5090 (1991).
[CrossRef] [PubMed]

Abramochkin, E. G.

E. G. Abramochkin and V. G. Volostnikov, J. Opt. A 6, S157 (2004).
[CrossRef]

Allen, L.

G. Nienhuis and L. Allen, Phys. Rev. A 48, 656 (1993).
[CrossRef] [PubMed]

Battipede, F.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, Phys. Rev. A 43, 5090 (1991).
[CrossRef] [PubMed]

Brambilla, M.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, Phys. Rev. A 43, 5090 (1991).
[CrossRef] [PubMed]

Cabrera, E.

E. Cabrera, O. G. Calderón, S. Melle, and J. M. Guerra, Phys. Rev. A 73, 053820 (2006).
[CrossRef]

Calderón, O. G.

E. Cabrera, O. G. Calderón, S. Melle, and J. M. Guerra, Phys. Rev. A 73, 053820 (2006).
[CrossRef]

Chen, Y. F.

T. H. Lu, Y. C. Lin, Y. F. Chen, and K. F. Huang, Phys. Rev. Lett. 101, 233901 (2008).
[CrossRef] [PubMed]

Y. F. Chen, T. H. Lu, K. W. Su, and K. F. Huang, Phys. Rev. Lett. 96, 213902 (2006).
[CrossRef] [PubMed]

K. F. Huang, Y. F. Chen, H. C. Lai, and Y. P. Lan, Phys. Rev. Lett. 89, 224102 (2002).
[CrossRef] [PubMed]

Dangoisse, D.

D. Dangoisse, D. Hennequin, C. Lepers, E. Louvergneaux, and P. Glorieux, Phys. Rev. A 46, 5955 (1992).
[CrossRef] [PubMed]

Glorieux, P.

D. Dangoisse, D. Hennequin, C. Lepers, E. Louvergneaux, and P. Glorieux, Phys. Rev. A 46, 5955 (1992).
[CrossRef] [PubMed]

Gorshkov, V. N.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, Phys. Rev. A 56, 4064 (1997).
[CrossRef]

Guerra, J. M.

E. Cabrera, O. G. Calderón, S. Melle, and J. M. Guerra, Phys. Rev. A 73, 053820 (2006).
[CrossRef]

Heckenberg, N. R.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, Phys. Rev. A 56, 4064 (1997).
[CrossRef]

Hennequin, D.

D. Dangoisse, D. Hennequin, C. Lepers, E. Louvergneaux, and P. Glorieux, Phys. Rev. A 46, 5955 (1992).
[CrossRef] [PubMed]

Huang, K. F.

T. H. Lu, Y. C. Lin, Y. F. Chen, and K. F. Huang, Phys. Rev. Lett. 101, 233901 (2008).
[CrossRef] [PubMed]

Y. F. Chen, T. H. Lu, K. W. Su, and K. F. Huang, Phys. Rev. Lett. 96, 213902 (2006).
[CrossRef] [PubMed]

K. F. Huang, Y. F. Chen, H. C. Lai, and Y. P. Lan, Phys. Rev. Lett. 89, 224102 (2002).
[CrossRef] [PubMed]

Lai, H. C.

K. F. Huang, Y. F. Chen, H. C. Lai, and Y. P. Lan, Phys. Rev. Lett. 89, 224102 (2002).
[CrossRef] [PubMed]

Lan, Y. P.

K. F. Huang, Y. F. Chen, H. C. Lai, and Y. P. Lan, Phys. Rev. Lett. 89, 224102 (2002).
[CrossRef] [PubMed]

Lepers, C.

D. Dangoisse, D. Hennequin, C. Lepers, E. Louvergneaux, and P. Glorieux, Phys. Rev. A 46, 5955 (1992).
[CrossRef] [PubMed]

Lin, Y. C.

T. H. Lu, Y. C. Lin, Y. F. Chen, and K. F. Huang, Phys. Rev. Lett. 101, 233901 (2008).
[CrossRef] [PubMed]

Louvergneaux, E.

D. Dangoisse, D. Hennequin, C. Lepers, E. Louvergneaux, and P. Glorieux, Phys. Rev. A 46, 5955 (1992).
[CrossRef] [PubMed]

Lu, T. H.

T. H. Lu, Y. C. Lin, Y. F. Chen, and K. F. Huang, Phys. Rev. Lett. 101, 233901 (2008).
[CrossRef] [PubMed]

Y. F. Chen, T. H. Lu, K. W. Su, and K. F. Huang, Phys. Rev. Lett. 96, 213902 (2006).
[CrossRef] [PubMed]

Lugiato, L. A.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, Phys. Rev. A 43, 5090 (1991).
[CrossRef] [PubMed]

Malos, J. T.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, Phys. Rev. A 56, 4064 (1997).
[CrossRef]

Melle, S.

E. Cabrera, O. G. Calderón, S. Melle, and J. M. Guerra, Phys. Rev. A 73, 053820 (2006).
[CrossRef]

Nienhuis, G.

G. Nienhuis and J. Visser, J. Opt. A 6, 248 (2004).
[CrossRef]

J. Visser and G. Nienhuis, Phys. Rev. A 70, 013809 (2004).
[CrossRef]

S. J. Van Enk and G. Nienhuis, J. Mod. Opt. 41, 963 (1994).
[CrossRef]

G. Nienhuis and L. Allen, Phys. Rev. A 48, 656 (1993).
[CrossRef] [PubMed]

S. J. van Enk and G. Nienhuis, Opt. Commun. 94, 147 (1992).
[CrossRef]

Penna, V.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, Phys. Rev. A 43, 5090 (1991).
[CrossRef] [PubMed]

Prati, F.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, Phys. Rev. A 43, 5090 (1991).
[CrossRef] [PubMed]

Schwinger, J.

J. Schwinger, in Quantum Theory of Angular Momentum, L.C.Biedenharn and H.van Dam, eds. (Academic, 1965), pp. 229-279.

Soskin, M. S.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, Phys. Rev. A 56, 4064 (1997).
[CrossRef]

Su, K. W.

Y. F. Chen, T. H. Lu, K. W. Su, and K. F. Huang, Phys. Rev. Lett. 96, 213902 (2006).
[CrossRef] [PubMed]

Tamm, C.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, Phys. Rev. A 43, 5090 (1991).
[CrossRef] [PubMed]

Van Enk, S. J.

S. J. Van Enk and G. Nienhuis, J. Mod. Opt. 41, 963 (1994).
[CrossRef]

S. J. van Enk and G. Nienhuis, Opt. Commun. 94, 147 (1992).
[CrossRef]

Vasnetsov, M. V.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, Phys. Rev. A 56, 4064 (1997).
[CrossRef]

Vicent, L. E.

Visser, J.

J. Visser and G. Nienhuis, Phys. Rev. A 70, 013809 (2004).
[CrossRef]

G. Nienhuis and J. Visser, J. Opt. A 6, 248 (2004).
[CrossRef]

Volostnikov, V. G.

E. G. Abramochkin and V. G. Volostnikov, J. Opt. A 6, S157 (2004).
[CrossRef]

Weiss, C. O.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, Phys. Rev. A 43, 5090 (1991).
[CrossRef] [PubMed]

Wolf, K. B.

J. Mod. Opt. (1)

S. J. Van Enk and G. Nienhuis, J. Mod. Opt. 41, 963 (1994).
[CrossRef]

J. Opt. A (2)

G. Nienhuis and J. Visser, J. Opt. A 6, 248 (2004).
[CrossRef]

E. G. Abramochkin and V. G. Volostnikov, J. Opt. A 6, S157 (2004).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

S. J. van Enk and G. Nienhuis, Opt. Commun. 94, 147 (1992).
[CrossRef]

Phys. Rev. A (6)

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, Phys. Rev. A 56, 4064 (1997).
[CrossRef]

G. Nienhuis and L. Allen, Phys. Rev. A 48, 656 (1993).
[CrossRef] [PubMed]

J. Visser and G. Nienhuis, Phys. Rev. A 70, 013809 (2004).
[CrossRef]

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, Phys. Rev. A 43, 5090 (1991).
[CrossRef] [PubMed]

D. Dangoisse, D. Hennequin, C. Lepers, E. Louvergneaux, and P. Glorieux, Phys. Rev. A 46, 5955 (1992).
[CrossRef] [PubMed]

E. Cabrera, O. G. Calderón, S. Melle, and J. M. Guerra, Phys. Rev. A 73, 053820 (2006).
[CrossRef]

Phys. Rev. Lett. (3)

K. F. Huang, Y. F. Chen, H. C. Lai, and Y. P. Lan, Phys. Rev. Lett. 89, 224102 (2002).
[CrossRef] [PubMed]

Y. F. Chen, T. H. Lu, K. W. Su, and K. F. Huang, Phys. Rev. Lett. 96, 213902 (2006).
[CrossRef] [PubMed]

T. H. Lu, Y. C. Lin, Y. F. Chen, and K. F. Huang, Phys. Rev. Lett. 101, 233901 (2008).
[CrossRef] [PubMed]

Other (1)

J. Schwinger, in Quantum Theory of Angular Momentum, L.C.Biedenharn and H.van Dam, eds. (Academic, 1965), pp. 229-279.

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Figures (5)

Fig. 1
Fig. 1

Experimental setup for the generation of high-order lasing modes in astigmatic cavities with an off-axis pumping scheme.

Fig. 2
Fig. 2

Experimental far-field patterns: (a) Lissajous pattern; (b) trochoidal pattern; (c), (d) other typical lasing patterns.

Fig. 3
Fig. 3

Numerical wave patterns for the intensity of eigenstates | Φ m , n α , β with ( m , n ) = ( 18 , 55 ) , α = π 2 : (a) β = 0 , (b) β = π 8 , (c) β = π 4 , (d) β = 3 π 8 , (e) β = π 2 .

Fig. 4
Fig. 4

(a)–(e) Geometric curves with Eq. (3) corresponding to the experimental wave patterns shown in Fig. 1. Detailed description for the parameters; see text.

Fig. 5
Fig. 5

(a) Propagation-dependent experimental patterns of Fig. 2a from beam waist to 0.33 z R ; (b) numerical wave patterns corresponding to the experimental wave patterns shown in Fig. 2c with ( p , q ) = ( 1 , 4 ) , ( m 0 , n 0 ) = ( 10 , 60 ) ; (c) phase distribution of the boxed region shown in (b).

Equations (3)

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Φ m , n ( HG ) ( x , y , z ) = 1 2 m + n 1 π m ! n ! 1 w ( z ) e ( x 2 + y 2 w ( z ) 2 ) H m ( 2 w ( z ) x ) H n ( 2 w ( z ) y ) ,
d s ( m + n ) 2 , m n 2 m + n 2 ( β ) = s ! ( m + n s ) ! m ! n ! ν = max [ 0 , s m ] min [ n , s ] ( 1 ) ν ( cos ( β 2 ) ) n + s 2 ν ( sin ( β 2 ) ) m s + 2 ν ν ! ( n ν ) ! ( s ν ) ! ( m s + ν ) ! .
[ X ̃ ( ϑ , z ) Y ̃ ( ϑ , z ) ] = [ e i α 2 cos ( β 2 ) e i α 2 sin ( β 2 ) e i α 2 sin ( β 2 ) e i α 2 cos ( β 2 ) ] [ X ( ϑ , z ) Y ( ϑ , z ) ] .

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